2017 AIMMS MOPTA competition
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1 2017 AIMMS MOPTA competition Production and Delivery of Radio-Pharmaceuticals to Medical Imaging Centers Team apio Advisor: Camilo Gómez, Ph.D Mariana Escallón (M.Sc student) Daniel López (M.Sc student) Santiago Ramírez (Leader, M.Sc student) Center for the Applied Optimization and Probability (COPA) Departamento de Ingeniería Industrial Universidad de los Andes (Colombia) 1
2 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 2
3 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 3
4 1. Problem description Production and distribution of radiopharmaceuticals (RP) A production center (PC) Imaging centers where RP are consumed PC :00 08:30 09:00 07:00 11:30 10:00 09:30 10:20 14:00 15:30 12:00 4
5 1. Problem description 5
6 1. Problem description 5
7 1. Problem description 5
8 1. Problem description 5
9 1. Problem description 5
10 1. Problem description Production problem 5
11 1. Problem description Production problem Production lines used 5
12 1. Problem description Production problem Production lines used Number of batches 5
13 1. Problem description Production problem Production lines used Number of batches Production time 5
14 1. Problem description Distribution problem 5
15 1. Problem description Distribution problem Definition of trips 5
16 1. Problem description Distribution problem Definition of trips Problem size 5
17 1. Problem description Distribution problem Definition of trips Problem size Which trips are assigned to a vehicle 5
18 1. Problem description Production problem Production lines used Number of batches Production time Distribution problem Definition of trips Problem size Which trips are assigned to a vehicle 5
19 1. Problem description Production problem Production lines used Number of batches Production time Relationship Distribution problem Definition of trips Problem size Which trips are assigned to a vehicle 5
20 1. Problem description - Production The PC has up to PL production lines. Each line has a number of dosages produced, and radioactivity level. different production time, PC PL1 PL2 PL3 PL4 } } } } Production time (min): 15 Number of dosages: 150 Radioactivity level (mci): 60 Production time (min): 30 Number of dosages: 100 Radioactivity level (mci): 120 Production time (min): 60 Number of dosages: 80 Radioactivity level (mci): 250 Production time (min): 120 Number of dosages: 60 Radioactivity level (mci):
21 1. Problem description - Distribution There are up to V vehicles for distribution. Each vehicle can return and depart multiple times from the PC and deliver multiple times to an imaging center a 4 PC 5 Route 1 vehicle a Route 2 vehicle a Route 3 vehicle a 7
22 1. Problem description - Distribution Dosages have to arrive at least 30 minutes before the appointment and with enough radioactivity tl k (lifespan) Unloading time 30 minutes Available to use 07:00 8
23 1. Problem description - Costs The model seeks to minimize the overall cost of production and distribution of one day, as well as an additional cost related to the unmet demand Production costs Fixed: ct Variable: cd Transportation costs Fixed: mp Variable: mt, mv Unattended demand M 9
24 1. Problem description Costs If an RP dose is not available, the examination is rescheduled, incurring in a cost M (per occurrence) Life of a patient Reputation of the imaging center Logistic costs 10
25 1. Problem description Costs If an RP dose is not available, the examination is rescheduled, incurring in a cost M (per occurrence) M 0 M
26 1. Problem description Costs If an RP dose is not available, the examination is rescheduled, incurring in a cost M (per occurrence) M 0 M 0 No production since we are minimizing costs 10
27 1. Problem description Costs If an RP dose is not available, the examination is rescheduled, incurring in a cost M (per occurrence) M 0 M 0 Start producing 10
28 1. Problem description Costs If an RP dose is not available, the examination is rescheduled, incurring in a cost M (per occurrence) M 0 M 0 Computationally complex implementation Trade off between producing and not servicing appointments 10
29 1. Problem description Costs If an RP dose is not available, the examination is rescheduled, incurring in a cost M (per occurrence) M 0 M 0 Service all appointments Feasible? 10
30 1. Problem description Costs If an RP dose is not available, the examination is rescheduled, incurring in a cost M (per occurrence) M 0 M 0 Sensitivity Analysis on M 10
31 1. Problem description Goal Minimize the production and distribution cost of RP. The value of M will significantly affect the solution. Study the effect of M on production and distribution of RP. 11
32 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 12
33 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production End 13
34 2. Solution strategy Set initial production MIP and additional constraint yes Stopping criterion no Optimize distribution Optimize production End 13
35 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production MIP problem End 13
36 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production Column generation approach End 13
37 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production One iteration End 13
38 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production Columns from previous iterations are used End 13
39 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production Columns from previous iterations are used End Solution improves or remains constant 13
40 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 14
41 2. Solution strategy Production problem Set initial production yes Stopping criterion no Optimize distribution Optimize production End 14
42 2. Solution strategy Production problem MIP fed by results from the distribution phase Production schedule Departure time from the production center of the vehicle in which the appointment is satisfied. Optimize production 15
43 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 16
44 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 Set of production lines (PL) 16
45 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 Set of minutes in planning horizon (T) 16
46 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 Set of appointments (C) 16
47 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 Hour of the appointment Hour in which the vehicle that supplies the appointment departs from the PC 16
48 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 Lifespan Number of units Production time 16
49 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 yb kt : 1 if production line k PL finishes a batch at t T, 0 otherwise y k : 1 if production line k PL is used at least once, 0 otherwise x ktl : 1 if production line k PL that ended its production at t T supplies demand of appointment l C 17
50 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 1 yb kt t 1 yb kt t =t tp k min T+tp k yb kt 1 t =min T k PL, t T t > tp k k PL Only one batch is produced at once for each production line 18
51 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 k PL,t T x ktl 1 l C Only one dosage is delivered to each appointment 19
52 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 N y k i C yb kt t T x kti b k yb kt k PL k PL, t T The number of dosages produced must not be exceeded 20
53 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 x ktl 0,1 k PL, t T, l C t + tl k ta l t td n Variables domain 21
54 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 t + tl k = 4: 00pm x ktl 0,1 k PL, t T, l C t + tl k ta l t td n 5: 00pm Variables domain 1. Lifespan 21
55 2. Solution strategy Production problem PC PL1 PL2 PL3 PL4 t = 11: 00am x ktl 0,1 td n = 10: 00am k PL, t T, l C t + tl k ta l t td n Variables domain 1. Lifespan 2. Appointment time 21
56 2. Solution strategy Production problem PC Objective function Production costs PL1 PL3 PL2 PL4 Unattended demand M Fixed: ct Variable: cd min M C x ptl + ct y k + cd l C,t T,k P k P k P,t T yb kt tp k 22
57 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 23
58 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production End 24
59 2. Solution strategy Distribution problem Column Generation Master Problem (MP) Auxiliary Problem (AP) 25
60 2. Solution strategy Distribution problem Column Generation Master Problem (MP) Auxiliary Problem (AP) Trip 1 Trip 2 Trip
61 2. Solution strategy Distribution problem Column Generation Master Problem (MP) Auxiliary Problem (AP) Trip 1 Trip 2 Trip
62 2. Solution strategy Distribution problem Column Generation Master Problem (MP) Auxiliary Problem (AP) Which appointments could be satisfied? Trip 1 Trip 2 Trip
63 2. Solution strategy Distribution problem Column Generation Master Problem (MP) Auxiliary Problem (AP) Trip 1 Trip 2 Trip
64 2. Solution strategy Distribution problem Column Generation Master Problem (MP) Auxiliary Problem (AP) Trip 1 Trip 2 Trip 3 Petal MC Appointments ,15,20 5,8 Which appointments will be satisfied? 25
65 2. Solution strategy Distribution problem Column Generation Master Problem (MP) Auxiliary Problem (AP) Trip 1 Trip 2 Trip 3 Petal MC 4 4 Appointments 10,15,20 5,8,9,11 Trip 1 Trip 2 Trip
66 2. Solution strategy Distribution problem (AP) i PC j 26
67 2. Solution strategy Distribution problem (AP) i PC Set of production batches (B) j 26
68 2. Solution strategy Distribution problem (AP) i PC Time at which batch is ready Lifespan j 26
69 2. Solution strategy Distribution problem (AP) i Set of medical centers (N) PC j 26
70 2. Solution strategy Distribution problem (AP) i Distance Time PC j 26
71 2. Solution strategy Distribution problem (AP) i PC Set of appointments (C) j 26
72 2. Solution strategy Distribution problem (AP) i PC Time of appointment j 26
73 2. Solution strategy Distribution problem (AP) i j Key variables z ij : 1 if trip travels from i N to j N, 0 otherwise h i : arrival time to i N w bl : 1 if trip can deliver units from batch b B to appointment l C (i.e., 1 if ta l hb b + ls b ), 0 otherwise d: departure time of the trip from the Production Center (MC 0) f b : 1 if trip can deliver units from batch b B, 0 otherwise r l : 1 if trip can deliver units to client l C, 0 otherwise 27
74 2. Solution strategy Distribution problem (AP) i j j N z ij j N z ji = 0 i N j N z ij 1 i N 28
75 2. Solution strategy Distribution problem (AP) i j z ij z ji = 0 j N j N j N z ij 1 i N i N Balance equations 28
76 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 h i + T ij + s i h j + 1 z ij T i N, j N i > 0 d h i T 0i z 0i + 1 z 0i T i N i >
77 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 h i + T ij + s i h j + 1 z ij T i N, j N i > 0 Arrival time to each medical center d h i T 0i z 0i + 1 z 0i T i N i >
78 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 h i + T ij + s i h j + 1 z ij T i N, j N i > 0 d h i T 0i z 0i + 1 z 0i T i N i > 0 Hour of departure from the PC 29
79 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 w bl 1 2 f b + r l d hb b 1 f b 1 T ta l h CMl s CMl 31 r l 1 T b B, l C ta l hb b + ls b b B l C 30
80 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 w bl 1 2 f b + r l d hb b 1 f b 1 T ta l h CMl s CMl 31 r l 1 T b B, l C ta l hb b + ls b b B l C Defines if an appointment could be served by the trip 30
81 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 w bl 1 2 f b + r l d hb b 1 f b 1 T ta l h CMl s CMl 31 r l 1 T b B, l C ta l hb b + ls b b B l C Defines if an appointment could be served by the trip 30
82 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 w bl 1 2 f b + r l b B, l C ta l hb b + ls b d hb b 1 f b 1 T ta l h CMl s CMl 31 r l 1 T b B l C Production occurs before vehicle departure time 30
83 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 w bl 1 2 f b + r l d hb b 1 f b 1 T ta l h CMl s CMl 31 r l 1 T b B, l C ta l hb b + ls b b B l C Appointment time occurs after vehicle arrival time to the Medical Center 30
84 2. Solution strategy Distribution problem (AP) h i i h j j d T 0i T ij T j0 h 0 w bl 1 2 f b + r l d hb b 1 f b 1 T ta l h CMl s CMl 31 r l 1 T b B, l C ta l hb b + ls b b B l C Defines if an appointment could be served by the trip 30
85 2. Solution strategy Distribution problem (AP) Objective function Reduced cost of Master Problem Column costs Time and distance Dual variables (MP) 31
86 2. Solution strategy Distribution Problem (AP) Objective function Reduced cost of Master Problem Column costs Time and distance Dual variables (MP) min mt 60 h o d + i N j N mv L ij z ij t T α t a t b B l C β bl w bl 31
87 2. Solution strategy Distribution Problem (AP) Objective function Reduced cost of Master Problem Column costs Time and distance Dual variables (MP) min mt 60 h o d + i N j N mv L ij z ij t T α t a t b B l C β bl w bl 31
88 2. Solution strategy Distribution Problem (AP) Objective function Reduced cost of Master Problem Column costs Time and distance Dual variables (MP) min mt 60 h o d + i N j N mv L ij z ij t T α t a t b B l C β bl w bl 31
89 2. Solution strategy Distribution Problem Column Generation Master problem (MP) Auxiliary problem (AP) Trip 1 Trip 2 Trip 3 Petal MC 4 4 Appointments 10,15,20 5,8,9,11 Trip 1 Trip 2 Trip
90 2. Solution strategy Distribution Problem (MP) PC PC
91 2. Solution strategy Distribution Problem (MP) PC PC 5 Set of production batches (B)
92 2. Solution strategy Distribution Problem (MP) PC PC 5 Units produced Time in which batch is ready Units delivered to PC
93 2. Solution strategy Distribution Problem (MP) PC PC Set of medical centers (N) 33
94 2. Solution strategy Distribution Problem (MP) PC PC Set of appointments (C) 33
95 2. Solution strategy Distribution Problem (MP) PC PC Time of the appointment 33
96 2. Solution strategy Distribution Problem (MP) PC PC Set of trips (Ω) 33
97 2. Solution strategy Distribution Problem (MP) PC PC Time at which a trip is used Appointments that could be satisfied by the trip 33
98 2. Solution strategy Distribution Problem (MP) PC PC x ω : 1 if trip ω Ω is used, 0 otherwise y: number of vehicles used v bk : 1 if demand of customer k C is satisfied by batch b B, 0 otherwise 34
99 2. Solution strategy Distribution Problem (MP) b B v bk 1 k C k C v bk u b ucm b b B ω Ω a tω x ω y y V t T 35
100 2. Solution strategy Distribution Problem (MP) b B v bk 1 k C One dosage delivered to a patient k C v bk u b ucm b b B ω Ω a tω x ω y y V t T 35
101 2. Solution strategy Distribution Problem (MP) b B k C v bk 1 k C v bk u b ucm b b B The number of dosages produced must not be exceeded ω Ω a tω x ω y y V t T 35
102 2. Solution strategy Distribution Problem (MP) b B v bk 1 k C k C v bk u b ucm b b B ω Ω a tω x ω y y V t T A maximum of V vehicles can be used 35
103 2. Solution strategy Distribution Problem (MP) ω Ω a tω x ω y y V t T A maximum of V vehicles can be used 36
104 2. Solution strategy Distribution Problem (MP) 10 a.m. 7 y ω Ω a tω x ω y y V t T A maximum of V vehicles can be used 36
105 2. Solution strategy Distribution Problem (MP) 10 a.m. 5 y ω Ω a tω x ω y y V t T A maximum of V vehicles can be used 36
106 2. Solution strategy Distribution Problem (MP) ω Ω θ k hb b +ls b w ωbk x ω v bk b B, k C If an appointment will be satisfied 37
107 2. Solution strategy Distribution Problem (MP) ω Ω θ k hb b +ls b w ωbk x ω v bk b B, k C If an appointment will be satisfied 37
108 2. Solution strategy Distribution Problem (MP) ω Ω θ k hb b +ls b w ωbk x ω v bk b B, k C If an appointment will be satisfied 37
109 2. Solution strategy Distribution Problem (MP) Objective function Minimized costs Column costs Vehicles costs Unattended demand min c ω x ω + mf y + M C M ω Ω b B k C v bk 38
110 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production End 39
111 2. Solution strategy Distribution Problem Initial artificial columns Optimize relaxed MP New column End Dual values yes Optimize AP Objective function<0* no Optimize MP x ω {0,1} 40
112 2. Solution strategy Distribution Problem Initial artificial columns Optimize relaxed MP New column Dual values yes Optimize AP Objective function<0* First integer solution with Z < 0 no End Optimize MP x ω {0,1} 40
113 2. Solution strategy Distribution Problem Initial artificial columns Optimize relaxed MP New column Dual values yes Optimize AP Objective function<0* First integer solution with Z < 0 no End Optimize MP x ω {0,1} Reduce computational time 40
114 2. Solution strategy Distribution Problem Initial artificial columns Optimize relaxed MP New column Dual values yes Optimize AP Objective function<0* Maximum of 20 columns no End Optimize MP x ω {0,1} 40
115 2. Solution strategy Distribution Problem Initial artificial columns Optimize relaxed MP New column Dual values yes Optimize AP Objective function<0* Maximum of 20 columns no End Optimize MP x ω {0,1} Reduce computational time 40
116 2. Solution strategy Distribution Problem Initial artificial columns Optimize relaxed MP New column Dual values yes Optimize AP Objective function<0* Strategy for column diversification no End Optimize MP x ω {0,1} 40
117 2. Solution strategy Distribution Problem Initial artificial columns Optimize relaxed MP New column Dual values yes Optimize AP Objective function<0* Strategy for column diversification no End Optimize MP x ω {0,1} Maximum number of MC 40
118 2. Solution strategy Distribution Problem Initial artificial columns Optimize relaxed MP New column End Dual values yes Optimize AP Objective function<0* no Optimize MP x ω {0,1} 40
119 2. Solution strategy Distribution Problem Initial artificial columns Solution of the production problem Optimize relaxed MP New column Dual values yes Optimize AP Objective function<0* Batch hb k ls k (mins) b_k (units) no End Optimize MP x ω {0,1} 40
120 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 41
121 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production End 42
122 2. Solution strategy Set initial production yes Stopping criterion no Optimize distribution Optimize production Columns from previous iterations are used End Solution improves or remains constant 42
123 2. Solution strategy Stopping condition Total costs + + Unattended demand Production Distribution 43
124 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 44
125 3. Stochastic approach How would the modelling approach change if the traveling time between imaging centers is stochastic? 45
126 3. Stochastic approach How would the modelling approach change if the traveling time between imaging centers is stochastic? (1.5T ij, T ij, 0.9T ij with 1/3 probability each) 45
127 3. Stochastic approach How would the modelling approach change if the traveling time between imaging centers is stochastic? (1.5T ij, T ij, 0.9T ij with 1/3 probability each) Pessimist solution 1.5T ij Intermediate solution T ij Average solution T ij Optimist solution 0.9T ij 45
128 3. Stochastic approach Pessimist solution 1.5T ij Intermediate solution T ij Average solution T ij Optimist solution 0.9T ij 46
129 3. Stochastic approach Pessimist solution 1.5T ij Intermediate solution T ij Average solution T ij Optimist solution 0.9T ij Monte Carlo Simulation for traveling times 46
130 3. Stochastic approach Monte Carlo Simulation for traveling times PC 47
131 3. Stochastic approach Monte Carlo Simulation for traveling times PC 1.5T ij 47
132 3. Stochastic approach Monte Carlo Simulation for traveling times T ij 1.5T ij 4 PC T ij 1.5T ij 47
133 3. Stochastic approach Monte Carlo Simulation for traveling times? T ij 1.5T ij 4 PC T ij 1.5T ij 47
134 3. Stochastic approach Monte Carlo Simulation for traveling times? T ij 1.5T ij 4 PC T ij 1.5T ij 47
135 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 48
136 4. AIMMS user interface and results 12: 00 am 6: 30 pm 6 vehicles available 7 medical centers + production center 315 appointments 4 production lines 49
137 4. AIMMS user interface and results 50
138 4. AIMMS user interface and results 50
139 4. AIMMS user interface and results 51
140 4. AIMMS user interface and results 51
141 4. AIMMS user interface and results 52
142 4. AIMMS user interface and results 53
143 4. AIMMS user interface and results 54
144 Cost (USD) 4. AIMMS user interface and results (M=200) Rescheduling Cost Production Cost Distribution Cost Total Cost Iteration 55
145 Total cost (USD) 4. AIMMS user interface and results (M=200) Iteration 56
146 Total cost (USD) 4. AIMMS user interface and results (M=200) Total cost: 17,431 Computational time (min): Unattended demand: Iteration 56
147 4. AIMMS user interface and results 57
148 4. AIMMS user interface and results 57
149 4. AIMMS user interface and results 57
150 4. AIMMS user interface and results 57
151 4. AIMMS user interface and results 58
152 4. AIMMS user interface and results 58
153 4. AIMMS user interface and results 58
154 4. AIMMS user interface and results 58
155 4. AIMMS user interface and results 58
156 4. AIMMS user interface and results 58
157 4. AIMMS user interface and results 58
158 4. AIMMS user interface and results 58
159 4. AIMMS user interface and results 59
160 Cost (USD) 4. AIMMS user interface and results Rescheduling cost Production cost Distribution cost M (Rescheduling cost) 60
161 Attended appointments 4. AIMMS user interface and results M (Rescheduling cost) 61
162 Deliveries Deliveries 4. AIMMS user interface and results M (Rescheduling cost) M (Rescheduling cost) 61
163 Computational Time (min) 4. AIMMS user interface and results M (Rescheduling cost) 62
164 4. AIMMS user interface and results 63
165 4. AIMMS user interface and results 63
166 4. AIMMS user interface and results Pessimist solution 1.5T ij Intermediate solution T ij Average solution T ij Optimist solution 0.9T ij 64
167 Distribution cost 4. AIMMS user interface and results Stochastic and deterministic distribution costs Pessimist Optimist Intermediate Average Stochastic cost range Mean cost Deterministic cost 65
168 Unattended demand 4. AIMMS user interface and results 60 Stochastic and deterministic unattended demand Pessimist Optimist Intermediate Average Unattended demand range Average Deterministic 66
169 Agenda 1. Problem description 2. Solution strategy Production problem Distribution problem Stopping condition 3. Stochastic approach 4. AIMMS user interface and results 5. Conclusions 67
170 5. Conclusions Reduced computational time Improve objective function AIMMS user interface Stochastic implementation in AIMMS 68
171 5. Conclusions Reduced computational time Improve objective function AIMMS user interface Stochastic implementation in AIMMS An efficient framework was proposed in terms of computational time and results.. All appointments were satisfied 68
172 5. Conclusions. Production problem Reach optimality by reducing time set Column generation strategies were implemented to reduce computational time Distribution problem. Relationship Constant improvement in the objective function was achieved by saving information from previous iterations 69
173 Questions and answers Thank you! 70
174 Literature review Azi, N., Gendreau, M., & Potvin, J. (2010). An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles. European Journal of Operational Research, 202(3), Boudia, M., Dauzere-pe, S., Prins, C., & Louly, M. A. (2006). Integrated optimization of production and distribution for several products. Chen, H., Hsueh, C., & Chang, M. (2009). Computers & Operations Research Production scheduling and vehicle routing with time windows for perishable food products, 36,
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